Volume elastic modulus with exponential function of transmural pressure as a valid stiffness measure derived by photoplethysmographic volume-oscillometry in human finger and radial arteries: potential for arteriosclerosis screening

Abstract

Noninvasive and convenient measurement of vascular stiffness is of considerable importance for early detection and treatment of arteriosclerosis. Volume elastic modulus (${E}_{{v}}$) is one of representative measures reflecting effective vascular elasticity that is strongly dependent upon blood pressure (BP) or transmural pressure (${P}_{{t}{r}}$ = mean BP – (externally applied pressure)). However, its nonlinear nature in terms of functional form has not been fully investigated in human vasculature. This paper therefore seeks to clarify the functional form of ${E}_{{v}}({P}_{{t}{r}})$ in the human finger and radial arteries based on photoplethysmographic volume-oscillometry developed for novel indirect BP measurement. Using a smartphone-based instrument specially designed for this study, ${E}_{{v}}$ values at various ${P}_{{t}{r}}$ levels were obtained in 11 male and female volunteers with various ages. It was demonstrated that ${E}_{{v}}({P}_{{t}{r}})$ showed an exponential behavior with respect to ${P}_{tr}$ changes, expressed as ${E}_{{v}}({P}_{{t}{r}})={E}_{{v}0}\bullet {e}{x}{p}(\alpha \bullet {P}_{{t}{r}})$ (${E}_{{v}0}$, α; constant) with a high coefficient of determination, the validity of which was also supported through theoretical derivation. Conclusively, the ${E}_{{v}}({P}_{{t}{r}})$ is found to increase exponentially with arterial distending pressure, and the independent measures ${E}_{{v}0}$ and α would be useful parameters to conveniently evaluate progressive changes of vascular stiffness among and/or within individuals, indicating that this measurement has potential for arteriosclerosis screening (200/200).

Graphical abstract

Schematic diagram of overall configuration of the measurement system of arterial elasticity in the finger and the wrist, consisting of a measuring, signal processing and control (MSC) unit (surrounded by the dashed line) and a smartphone for data display and storage. An occlusive cuff and a photoplethysmographic placement of LED and PD for the finger and the wrist are shown in the upper middle part. Measurement scenes of the finger and the wrist are also inset in the upper left and in the upper right part, respectively.

Appendix

It is not easy to obtain directly the PWV in the radial artery just under the wrist cuff (${PWV}_{r}$) with high accuracy due to the considerable difficulty in measuring the pulse transit time ($PTT$) over a distance equivalent to the cuff width (70 mm in this case). We therefore measured the $PTT$ between the longer distance (about 200 mm or more) of a portion near the elbow and the distal end of the cuff, to obtain ${PWV}_{{r}}$.

Two reflectance-type $PPG$ sensors are used to detect the upstream (${PPG}_{{U}}$) and the downstream $PPG$ signal (${PPG}_{{D}}$), respectively, at the proximal portion near the elbow (U) and at the distal end of the cuff (D), as shown in Fig. 6(a)-(c). The LED and the PD were the same as described in the text and the distance between the centers of the LED and the PD element was 10 mm. The distance from “U” to the proximal end of the cuff and that from “U” to “D” denote, respectively, the symbols L₀ and L (cuff width $W=L-{L}_{0}$), and we assume that the $PWV$ of distance ${L}_{0}$ (${PWV}_{0}$) is constant during the application of ${P}_{{c}}$, that is, the change in $P_{{{\text{tr}}}} ~\left( { = MBP - P_{c} } \right)$. Let ${PTT}_{0}$ and ${PTT}_{{r}}$ be, respectively, the $PTT$ of distance ${L}_{0}$ and that of distance $W$ of the radial artery under the cuff, then the following equations can be written as:

$$\begin{array}{*{20}c} {PTT_{0} = L_{0} /PWV_{0} {\text{ ;}}} & {PTT_{{\text{r}}} \left( {P_{{{\text{tr}}}} } \right) = \left( {L - L_{0} } \right)/PWV_{{\text{r}}} \left( {P_{{{\text{tr}}}} } \right)} \\ \end{array}$$

(10)

The $PTT$ between “U” and ‘D’ can be obtained from the time difference between the rising points of the ${PPG}_{{U}}$ and the corresponding ${PPG}_{{D}}$ waveform, and expressed by:

$$PTT={PTT}_{0}+{PTT}_{{r}}$$

(11)

Averaged $PWV({PWV}_{{a}{v}{e}})$ between “U” and “D” can therefore be expressed as:

$$\begin{array}{*{20}c} {{{{{PWV_{{{\text{ave}}}} = L} \mathord{\left/ {\vphantom {{PWV_{{{\text{ave}}}} = L} {PTT = L}}} \right. \kern-\nulldelimiterspace} {PTT = L}}} \mathord{\left/ {\vphantom {{{{PWV_{{{\text{ave}}}} = L} \mathord{\left/ {\vphantom {{PWV_{{{\text{ave}}}} = L} {PTT = L}}} \right. \kern-\nulldelimiterspace} {PTT = L}}} {\left( {PTT_{0} + PTT_{{\text{r}}} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {PTT_{0} + PTT_{{\text{r}}} } \right)}}} \\ { = {L \mathord{\left/ {\vphantom {L {\left( {{{L_{0} /PWV_{0} + ~\left( {L - L_{0} } \right)} \mathord{\left/ {\vphantom {{L_{0} /PWV_{0} + ~\left( {L - L_{0} } \right)} {PWV_{{\text{r}}} \left( {P_{{{\text{tr}}}} } \right)}}} \right. \kern-\nulldelimiterspace} {PWV_{{\text{r}}} \left( {P_{{{\text{tr}}}} } \right)}}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{{L_{0} /PWV_{0} + ~\left( {L - L_{0} } \right)} \mathord{\left/ {\vphantom {{L_{0} /PWV_{0} + ~\left( {L - L_{0} } \right)} {PWV_{{\text{r}}} \left( {P_{{{\text{tr}}}} } \right)}}} \right. \kern-\nulldelimiterspace} {PWV_{{\text{r}}} \left( {P_{{{\text{tr}}}} } \right)}}} \right)}}} \\ \end{array}$$

(12)

Using this equation, ${PWV}_{{r}}({P}_{{t}{r}})$ can be obtained as:

$${PWV}_{{r}}\left({P}_{{t}{r}}\right)={PWV}_{0}\bullet {PWV}_{{a}{v}{e}}\bullet (L-{L}_{0})/({PWV}_{0}\bullet L-{PWV}_{{a}{v}{e}}\bullet {L}_{0})$$

(13)

When ${P}_{{t}{r}}=MBP$ (i.e., ${P}_{{c}}=0$), ${PWV}_{{a}{v}{e}}={PWV}_{0}$, and thus ${PWV}_{{r}}\left(MBP\right)={PWV}_{0}$ (see also Fig. 6a). Since each quantity on the right side of Eq. (13) is measurable, we can obtain the ${PWV}_{{r}}$ at various ${P}_{{t}{r}}$ levels. Therefore, ${E}_{{v}{r}{c}}$ can be calculated by Eq. (4).

The ${PPG}_{{U}}$ and ${PPG}_{{D}}$ signals detected by a $PPG$ device specially designed for this experiment were fed to a note book PC (Inspiration 11, 3000 Series; Dell Technologies Japan Inc., Kanagawa, Japan) via an AD converter (NI USB-6210, 16-bit; National Instruments Corp., Austin, USA) with 1 kHz sampling frequency. LabVIEW (National Instruments Corp., Austin, USA) was used to record the $PPG$ signals and to analyze the rising points of the signals with appropriate software. The $PPG$ measurement was simultaneously made with the ${E}_{v}$ measurement by the MSC unit.

Figure 7 shows example recordings of ${PPG}_{{U}}$ and ${PPG}_{{D}}$ at ${P}_{{c}}=0$ (upper record) and ${P}_{{c}}=40$ mmHg (lower record) obtained in one subject. The values of ${PWV}_{{r}}\left(0\right)$ and ${PWV}_{{r}}\left(40\right)$ calculated by Eq. (13) along with those of ${E}_{{v}{r}{c}}(0)$ and ${E}_{{v}{r}{c}}(40)$ by Eq. (4) are indicated in the figure caption. The simultaneous measurements were done at least 3 times to acquire the mean $PTT$ value, calculating ${PWV}_{{r}}\left({P}_{{t}{r}}\right)$ using Eq. (13). This experiment was carried out separately after the finger- and the wrist-${E}_{{v}}$ measurements were completely finished.

Fig. 6

Schematic drawings to explain how to obtain pulse wave velocity under the occlusive wrist cuff (${PWV}_{{c}}({P}_{{t}{r}}))$ at various ${P}_{{t}{r}}$ levels from the detection of an upstream (${PPG}_{{U}}$) and a downstream $PPG$ signal (${PPG}_{{D}}$) using $PPG$ sensors placed, respectively, at the proximal portion near the elbow (U) and at the distal end of the cuff (D). (a) is a state at ${P}_{{c}}=0$ (${P}_{{t}{r}}=MBP$), (b) 0 < ${P}_{{c}}$  < MBP, and (c) ${P}_{{c}}=MBP$. The other symbols used in this figure are as follows: L, distance between the portions “U” and “D”; L₀, distance between the portion “U” and the proximal end of the wrist cuff; and PWV₀, pulse wave velocity of the distance L₀

Fig. 7

Examples of simultaneous recordings of ${PPG}_{{U}}$ and ${PPG}_{{D}}$ at ${P}_{{c}}=0$ ((a); upper record) and ${P}_{{c}}=40$ mmHg ((b); lower record) obtained in one subject (L = 25 cm, ${L}_{0}=18$ cm). The rising points of both ${PPG}_{{U}}$ and ${PPG}_{{D}}$ records are indicated by black dots. The values of PTT at ${P}_{{c}}=0$ ($PTT(0)$) and that at ${P}_{{c}}=40$ ($PTT(40)$) are 29 and 53 ms, respectively, in this case, thus being ${PWV}_{{r}}\left(0\right)=8.62$ and ${PWV}_{{r}}\left(40\right)=2.81$ m/s by calculation from Eq. (13) and ${E}_{{v}{r}{c}}\left(0\right)=585$ and ${E}_{{v}{r}{c}}\left(40\right)=62$ mmHg by Eq. (4) as indicated in the inset